Speaker: Davar Khoshnevisan (University of Utah)

Title: Dissipation and High Disorder

Abstract: We consider the "parabolic Anderson model" with delta initial function. This is a linear, infinite system of stochastic differential equations that arise in a vast number of physical models. The solution of this system models, among other things, the particle density of of an infinite system of independent random walks, which replicate [give birth] to new random walks according to a common [independent] space-time white noise environment, starting with one particle at the origin at time 0. We show that, when the underlying random walks move in 1 or 2 dimensions, the total number of particles vanishes as time goes to infinity. By contrast, in dimensions 3 or greater there is phase transition: If the variance of the noise is sufficiently high, then the total number of particles vanishes; and if the noise variance is not sufficiently high, then the total number of particles tends to a non-trivial random variable. This talk is based on joint work with Le Chen, Michael Cranston, and Kunwoo Kim.